Problem: The geometric sequence $a_i$ is defined by the formula: $a_1 = -\dfrac13$ $a_i = a_{i - 1} \cdot (-3)$ Find the sum of the first $75$ terms in the sequence. Choose 1 answer: Choose 1 answer: (Choice A) A $-1.01\cdot10^{35}$ (Choice B) B $-6.76\cdot10^{34}$ (Choice C) C $ -5.06\cdot10^{34} $ (Choice D) D $1.69\cdot10^{34}$
Explanation: Getting started Let's write out the first few terms of the series: $-\dfrac13 + 1 -3...$ We're dealing with a geometric series because each term is multiplied by $-3$ to get the next term. We need a formula to compute the sum of the terms. Formula for geometric series The sum $S_n$ of a finite geometric series is $S_n = \dfrac{a_1(1-r^n)}{1-r}$ where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. What do we need to use the formula? The first term $(a_1 = {-\dfrac13})$ and the number of terms $(n = {75})$ are given in the question. The common ratio $r$ is ${-3}$ because each term is multiplied by ${-3}$ to get the next term. Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac{a_1(1-r^n)}{1-r} \\\\ S_{{75}}&=\dfrac{{-\dfrac13}(1-\left({-3}\right)^{{75}})}{1-\left({-3}\right)} \\\\ S_{{75}}&=-\dfrac1{12}(1-\left({-3}\right)^{{75}}) \\\\ S_{{{75}}} &\approx -5.06\cdot10^{34} \end{aligned}$ The answer $ -5.06\cdot10^{34} $